{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r03-distance-formula-0098",
  "pilot_number": "R0007-04048",
  "number": 4048,
  "family_id": "qf-density10-distance-integer-sum",
  "title": {
    "zh-Hant": "第 4048 題：兩段完全平方距離的整數和",
    "en": "Question 4048: Sum of Two Integer Principal Distances"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r03-distance-formula-0098",
    "batch_id": "batch-20261008-advance-20261010-r03-distance-formula-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-distance-formula"
    ],
    "title_zh": "兩段完全平方距離的整數和",
    "stem_zh": "同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[66, 88], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。",
    "answer": {
      "kind": "expression",
      "value": "123",
      "explanation_zh": "兩段整數距離和=123。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "兩段完全平方距離的整數和"
      ]
    },
    "variant": {
      "family_id": "qf-density10-distance-integer-sum",
      "seed": 202610103098,
      "parameters": {
        "vectors": [
          [
            66,
            88
          ],
          [
            5,
            12
          ]
        ],
        "frame": "orthogonal_equal_unit",
        "operation": "density10_distance_integer_sum",
        "domain": "integer_euclidean_plane",
        "semantic_template": "qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"66\",\"88\"]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r03-distance-formula-0098",
    "batch_id": "batch-20261008-advance-20261010-r03-distance-formula-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cd48011ff8fb622f3091c03de6e1f82dec2fb6263d14264c9696d1b57d685985",
      "family_id": "qf-density10-distance-integer-sum",
      "seed": 202610103098,
      "semantic_template": "qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"66\",\"88\"]]"
    },
    "fields": {
      "title": "Sum of Two Integer Principal Distances",
      "stem": "In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[66, 88], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances.",
      "answer": {
        "kind": "expression",
        "value": "123",
        "explanation": "Take each exact nonnegative integer square root before adding; a two-segment sum is not an endpoint straight-line distance. The exact result, retaining the requested order, is 123."
      },
      "skills": [
        "Sum of Two Integer Principal Distances"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[66, 88], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。",
        "en": "In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[66, 88], [5, 12]]; both squared norms are perfect squares. Find the sum of their nonnegative integer distances."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "integer vectors from a common origin;orthogonal equal-length units",
      "after": "L2(v)≥0 and L2(v)²=v₁²+v₂²",
      "relation": "implies",
      "rule_id": "euclidean_distance_frame",
      "conditions": [],
      "explanation": {
        "zh-Hant": "共同原點、正交及等單位是平方和公式的前提；向量座標可負，距離不可負。",
        "en": "A common origin,orthogonality and equal units support the sum-of-squares formula;coordinates may be negative but distances may not."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "vector1 coordinate1 squared := square(66)",
      "after": "vector1 coordinate1 squared = 4356",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector1 coordinate1 squared，不以小數近似替代。",
          "en": "Recompute vector1 coordinate1 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "66"
        ],
        "output": "4356",
        "label": "vector1 coordinate1 squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "vector1 coordinate2 squared := square(88)",
      "after": "vector1 coordinate2 squared = 7744",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector1 coordinate2 squared，不以小數近似替代。",
          "en": "Recompute vector1 coordinate2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "88"
        ],
        "output": "7744",
        "label": "vector1 coordinate2 squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "vector1 L2 squared := sum(4356, 7744)",
      "after": "vector1 L2 squared = 12100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個平方相加是距離平方，不是兩座標直接相加。",
        "en": "Add the coordinate squares to obtain squared distance,not a sum of the coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector1 L2 squared，不以小數近似替代。",
          "en": "Recompute vector1 L2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4356",
          "7744"
        ],
        "output": "12100",
        "label": "vector1 L2 squared"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "vector2 coordinate1 squared := square(5)",
      "after": "vector2 coordinate1 squared = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector2 coordinate1 squared，不以小數近似替代。",
          "en": "Recompute vector2 coordinate1 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "5"
        ],
        "output": "25",
        "label": "vector2 coordinate1 squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "vector2 coordinate2 squared := square(12)",
      "after": "vector2 coordinate2 squared = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector2 coordinate2 squared，不以小數近似替代。",
          "en": "Recompute vector2 coordinate2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "12"
        ],
        "output": "144",
        "label": "vector2 coordinate2 squared"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "vector2 L2 squared := sum(25, 144)",
      "after": "vector2 L2 squared = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個平方相加是距離平方，不是兩座標直接相加。",
        "en": "Add the coordinate squares to obtain squared distance,not a sum of the coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 vector2 L2 squared，不以小數近似替代。",
          "en": "Recompute vector2 L2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25",
          "144"
        ],
        "output": "169",
        "label": "vector2 L2 squared"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "principal root1 square := square(110)",
      "after": "principal root1 square = 12100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "核對整數主根的平方恰等原平方距離。",
        "en": "Check that the integer principal root squares to the original squared distance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal root1 square，不以小數近似替代。",
          "en": "Recompute principal root1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "110"
        ],
        "output": "12100",
        "label": "principal root1 square"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "root=110≥0;root²=12100=L2²",
      "after": "L2=110",
      "relation": "implies",
      "rule_id": "nonnegative_perfect_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負且平方正確才唯一確定距離；完全平方是本題前提，不以小數開根或四捨五入代替。",
        "en": "Nonnegativity and the correct square uniquely determine the distance;this exercise requires a perfect square,not a rounded decimal root."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "principal root2 square := square(13)",
      "after": "principal root2 square = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "核對整數主根的平方恰等原平方距離。",
        "en": "Check that the integer principal root squares to the original squared distance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 principal root2 square，不以小數近似替代。",
          "en": "Recompute principal root2 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "13"
        ],
        "output": "169",
        "label": "principal root2 square"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "root=13≥0;root²=169=L2²",
      "after": "L2=13",
      "relation": "implies",
      "rule_id": "nonnegative_perfect_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負且平方正確才唯一確定距離；完全平方是本題前提，不以小數開根或四捨五入代替。",
        "en": "Nonnegativity and the correct square uniquely determine the distance;this exercise requires a perfect square,not a rounded decimal root."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "two distance sum := sum(110, 13)",
      "after": "two distance sum = 123",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把兩個已核對的距離相加。",
        "en": "Add the two checked distances."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 two distance sum，不以小數近似替代。",
          "en": "Recompute two distance sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "110",
          "13"
        ],
        "output": "123",
        "label": "two distance sum"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "[110, 13]",
      "after": "sum=123",
      "relation": "implies",
      "rule_id": "sum_of_distances_not_endpoint_distance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是兩個從共同原點量得的長度之和，不擅自改成兩向量端點間的直線距離。",
        "en": "This sums two lengths measured from a common origin;do not reinterpret it as the straight-line distance between their endpoints."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "123",
    "principal_roots": [
      110,
      13
    ]
  },
  "conclusion": {
    "canonical_value": "123",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261008-advance-20261010-r03-distance-formula-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r03-distance-formula-foundations/questions.jsonl",
      "batch_sha256": "817c2ffbb228a708e250aa13865b5f5cc70e482a1c54d82a63d493bd6bed1630",
      "original_line": 98,
      "original_line_sha256": "838a0e8ae91ca2d311679d508e9bb40debe56108976692bf9767fd1b9fb676f2",
      "question_compact_sha256": "cd48011ff8fb622f3091c03de6e1f82dec2fb6263d14264c9696d1b57d685985",
      "english_row_compact_sha256": "ca2754a150bcd2ddec59dd0a1a9bb576b7b60c6857ffcfe89c91c2f446341574",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r03-distance-formula-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r03-distance-formula-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-distance-formula": {
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        "english_sha256": "d297d8ae3afb0c4aa0a2de4cbd9e17c6355f2ae3f0fbf1510b9dc7cea3b27827",
        "original_path": "public/data/mko/distance-formula.json",
        "english_path": "public/data/i18n/editions/mko-distance-formula.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "05192b67b85e5da371cc674c143f420297c46f335b5b399962862087d410d335",
    "original_candidate_line_sha256": "05192b67b85e5da371cc674c143f420297c46f335b5b399962862087d410d335"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p8",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 13,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "136c9bb1acd69271166e5644fe08242a488791749903cf35ffdc785cad1f0e56",
      "arithmetic_steps": 9
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}